FMS-2016-03-08

An Integrated Approach for Decision on Multi FMSs Combination Selection
Maryam DEHGHANBAGHI1
, Hasan HOSSEINI-NASAB2
, Jurgita ANTUCHEVICIENE3*
,
Ehsan MEHRABANFAR4
1
Department of Industrial Engineering
Yazd University, Iran
2
Department of Planning
Research Institute of Petroleum Industry (RIPI), Tehran, Iran
3
Department of Construction Engineering and Management
Vilnius Gediminas Technical University, Lithuania
4
Young Researchers and Elites Club, North Tehran Branch
Islamic Azad University, Tehran, Iran
e-mail: dehghanbaghim@ripi.ir, hhn@yazd.ac.ir, jurgita.antucheviciene@vgtu.lt,
e.mehrabanfar@aut.ac.ir
Abstract. In the fierce global competition, some objectives such as cost, quality and customer
satisfaction appears to be utmost significant. Flexible manufacturing systems (FMS) have a
great potential in manufacturing both cost effective and customer based products. These
systems bring us flexibility, but this flexibility accompanies with cost and time. Thus,
selecting suitable FMS necessitates excessive attention. The problem of FMS selection and
evaluation becomes more difficult when facing with multi FMSs selection problem. In this
paper, we propose an integrated approach to find a suitable combination of FMSs in a multi
FMSs decision making problem. Each FMS has several alternatives. Therefore, there are
many possible solutions for this problem. We first identify the objective and subjective
attributes. Second, grey system theory is applied to deal with the incomplete and uncertain
information of subjective data, and the objective data are extracted from simulation modeling.
A goal-programming model is then utilized to formulate the problem and to assign priorities
to the objectives. Finally, a genetic algorithm (FA) based model is applied to solve the
combination problem, as the formulated problem is difficult to be solved. The model proposed
in this paper determines the most appropriate FMSs combination and facilitates decision
making of such a hard problem.
Key words: flexible manufacturing systems (FMS); integrated approach; genetic algorithm
(GA); grey system theory; goal programming (GP)
1. Introduction & Literature Review
Ever changing customers’ needs lead managers to compete in the fierce global market. Thus,
flexible manufacturing systems (FMS) are used to take advantage of their competitive
preferences. These systems which include interconnected group of processing workstations
are called flexible since FMS is capable of processing a variety of different part styles
simultaneously at the various workstations (Rao, 2006). The concept of flexibility refers to
machines, processes, products, routings, volume or productions. One or more part families of
a product can utterly be processed on a continuous basis without human involvement by using
a generic FMS. It is flexible enough to cope with market fluctuations while requiring no new
equipment.
As a complex dynamic system, FMS is extremely important to optimize the
productivity by utilizing the available resources (Jyothi, 2012). Rao (2013) summarizes the
benefits of FMS in three basic achievements: increasing in product types, enhancement in
quality and reduction in WIP and setup costs. FMS aims to provide greater manufacturing
flexibility, less inventory and floor space, lower lead times, and longer useful life of

equipment over consecutive generations of different products (Kulak and Kahraman, 2005).
That is why FMSs have received great attention for over three decades. Due to increased
attention and investment on FMS technologies, the evaluation process for selecting the proper
technologies requires a logical systematic mathematical approach to make the best decision to
be conjugated with real requirements. FMS selection, as a complicate multi criteria decision
making problem, requires a robust decision support system to consider both monetary and
non-monetary criteria. Being multidimensional situations, these problems can be solved by
various techniques (Chakraborty et al., 2015). Karsak (2008), Khouja (1995) abd Liu (2008)
applied Data Envelopment Analysis (DEA) to evaluate and to select FMSs. Shang and
Sueyoshi (1995) proposed an evaluation procedure incorporating the AHP (analytical
hierarchy process), simulation, and DEA for the selection of FMS alternatives. In the last few
decades, a great number of researches have focused on the selection methodologies. We refer
the readers to (Chatterjee and Chakraborty, 2014) providing a review on advance
manufacturing technology selection based on MCDM techniques. Multi criteria decision
making (MCDM) tools are prevalent techniques used to justification of advanced
manufacturing technologies (AMTs) (Chuu, 2009; Karsak and Tolga, 2001). Stam and Kuula
(1991) proposed a two-phase decision model using AHP to reduce the number of alternatives
and provided a multi-objective mathematical programming model to select the most suitable
FMS.
An effective justification process requires the consideration of many quantitative
attributes and qualitative attributes. There are various attributes for the evaluation of
manufacturing technologies: cost, work in process (WIP), flexibility, quality, yield, floor
space, down time, number of employees, expandability, ease of use, competitiveness and so
on (Rao and Parnichkun, 2009; Sener and Karsak, 2007; Rao, 2006). Based on the literature
reviewed the attributes are classified into two categories: qualitative and quantitative (or
subjective & objective) (Chuu, 2009; Karsak and Tolga, 2001; Ragavan and Punniyamoorthy,
2003). The detailed attributes used in other studies are collected in Table 1. We can observe
that most attributes used nearly by all studies are flexibility, quality, cost (purchase & setup),
WIP, Yield or throughput and space required. The factors “Ease of use”, Reliability, “Labor
reduction”, capacity and learning are in less attention in the above mentioned studies.
Table 1
Detailed attributes based on the literature review.
Year and Authors
Flexibility
Quality
Cost
WIP
yield
Space
Required
DownTime
Labor
Reduction
Capacity
Easeofuse
learning
LeadTime
Reliability
TardyJob
2015 Chakraborty et al. * * * *
2014
Chatterjee and
Chakraborty
* * * * * *
2012 Mandal and Sarkar * * * *
2009 Rao and Parnichkun * * * * *
2010 Anvari et al. * * * * * *
2009 Chuu * * * * * * *
2008 Karsak * * * * * *
2008 Liu * * * * * *
2007 Sener and Karsak * * * * * *
2006 Rao * * * * * *
2002
Karsak and
Kuzgunkaya
* * * *
1998 Luong * * * * *
1995 Shang and Sueyoshi * * * * * *

Among these attributes, the qualitative factors (reliability, quality, etc.) obtained by
experts, are not crisp values and include vague and incomplete information. Also, collecting
exact data is a difficult task and seldom available.
Fuzzy mathematics and grey systems theory are two most-often applied theories
employed in studies of this kind of uncertainties. The main idea of fuzzy mathematics is based
on the membership functions established based on experiences. Thus, a number of previous
works on advanced manufacturing technologies’ (AMT) justification are based on fuzzy data.
Rao and Parnichkun (2009) proposed a fuzzy combinatorial mathematics-based decision-
making method for evaluation and ranking of FMS alternatives. Chuu (2009) applied fuzzy
multiple attributes group decision making with multiple fuzzy information to select the best
FMS alternative in Taiwanese bicycle industry. Karsak and Tolga (2001) developed a fuzzy
MCDM approach to evaluate advanced manufacturing system investments applying fuzzy
ranking method. Karsak and Kuzgunkaya (2002) proposed a multiple objective programming
approach based on fuzzy data to the selection of a proper FMS. Rao (2007) suggested a fuzzy
decision methodology based on digraph and matrix methods to facilitate the selection of a
suitable flexible manufacturing system. Liu (2008) presented a fuzzy DEA / assurance region
(AR) method capable of measuring the performance of FMS alternatives by crisp and fuzzy
data for the representation of inputs and outputs.
The grey systems theory, proposed by Deng (1985), is developed to study problems of
“small samples and poor information” and requires neither frequent distributions nor exact
membership functions. Also, grey theory -as the expansion of fuzzy theory (Jun, 1993) -
considers the condition of the fuzziness and deal flexibly with fuzzy situation (Tseng, 2008).
Applying this theory to the problem of FMSs selection has been seen in a few references. A
fuzzy MCDM approach based on the concept grey systems theory has been proposed by
Dhal et al., (2011) to systematically evaluate FMSs’ alternatives. In another research,
Samvedi et al. (2011) applied fuzzy AHP, grey relation analysis to rank the FMS’s tools.
To the best of our knowledge, among all these researches, there exists a lack of proper
solution when we have multi FMSs combination selection problem. In many manufacturing
companies such as automobile manufacturers, there are more than one FMS to be selected
when the sequencing is not a matter and when there are flexibility relations between the
alternatives of different FMSs items. In this study, we propose a GA based grey system theory
decision frame work to select the proper combination of multi FMSs in a factory. GA is
applied to select the near optimal solution as a quick response tool in this complicated
problem. Both qualitative and quantitative attributes are applied for this selection problem.
Each possible solution is assessed through a mathematical model using these attributes. The
attributes considered as objectives are: cost (investment and maintenance cost), decrease in
WIP (work in process), decrease in the number of tardy jobs, improvement in quality, space
required, reliability and Yield (as throughput minus scrape and rework). Since the qualitative
data accompanies incomplete information based on the experts’ scores and unavailability of
exact and crisp data and cannot be estimated by an exact numerical value, we use the Grey
systems theory in this selection problem.
The organization of the paper is as follows: Section 2 describes a complete description
on the multi FMS selection problem. In Section 3, the preliminaries of methodologies applied
to solve the problem are explained. Section 4 presents the proposed approach. In Section 5 an
illustrative example is developed to show the validity of the proposed model. Finally,
Section 6 contains a brief conclusion on the discussed issues.
2. Problem Descriptions
In the 21th
century, enterprises are approaching to the application of advanced
manufacturing technologies. There appears to be a few numbers of factories to be fully
automated meaning to manufacture the products without intervening of the human beings. In
some of these enterprises, like automobile manufacturers, there are more than one FMS to
produce one product since the different modules of the product have various manufacturing

technologies. For example, manufacturing the seats of an automobile completely differs from
its engine manufacturing process. Thus, they should have separate FMSs to manufacture.
These FMSs are interrelated due to their flexibility relationships. Therefore, devising a proper
combination of these FMSs necessitates when such decision-making appears. Each FMS has
its alternatives. Hence, if we have i.e. four FMSs each of which has five alternatives we have
4^5 possible solutions to find the best or even near optimum FMSs combination for the
factory as the four FMSs are in relation with each other with respect to their flexibility.
Therefore, finding a suitable alternative for each FMS, considering several objectives; cost,
quality, etc, in order to meet both customers and manufacturer requirements is a challenging
task. To reach the objective, providing a suitable model is necessary to assure the success of
the manufacturing process and to estimate the optimum FMSs combination. Thus, a set of
different combinations among the many possible solutions should be generated and the best
one selected in the minimum possible time to help decision makers in the decision process. In
this paper, an effective mathematical model based goal programming, solving by GA
technique, is proposed for the problem of FMS selection in an uncertain environment
considering both subjective and objective factors.
3. Preliminaries in Methodologies
3.1. Genetic Algorithm (GA)
The theory of Genetic Algorithms (GA) is initially proposed by Holland (1975) as a
probabilistic search method in an attempt to emulate Darwin theory of natural selection which
is the basic manifest of the most fitting survival. GA is operated to optimize a population of
candid solutions; each one presented by a string containing random numbers, by modification
the characteristics of the set of solutions. It is able to provide the capacity of sectioning the
optimum solution according to problem specifications, whether the criteria of concern are
nonlinear, constrained, multi-modal, discrete, or NP hard (Man et al., 1999).
The GA as a parallel processing mechanism takes searches simultaneously in multiple
starting points and for different areas, and since it is run based on continuous evolution of
generations it is able to increase the speed of finding an optimal solution (Lin et al., 1995). In
this regard, it is conclusive to apply GA in finding an optimal solution for multi FMS
selection in which the alternatives are interrelated enough to render overly difficult problems.
Conventional GA is operated stepwise based on some general rules to produce, stop,
and replicate generations. First of all some random numbers are produced and the gene serial
values are set to them to create the initial population. Then a fitness function is defined;
mainly acting as the performance indicator of GA it orders the fitness as the basic evaluation
criterion in GA, to the extent that higher a solution is fitted to the fitness function, it takes
more chance to survive in next generations. Then it is required to set a stop criterion, by using
the fixed number of generations the termination criterion is provided. The replication rule is
created to continue the iterative process; and finally crossover and mutation rates are
provided, one to acquire and one to avoid. Crossover procedure aims to acquire two
descendants with probability of crossover (PC). It initially pick up a pair of parent
chromosomes in a random manner from the current population, then performs single-point or
multi-point crossover, i.e. swap a share of the underlying data structure of the pair to breed the
progeny. Then this process is iterated to a point that there exist an appropriate number of
candidates in the next generation run. Mutation also as one of the fundamental GA operators
acts to avoid genetic similarity from one generation to the next. The purpose is to maintain the
required diversity; otherwise the evolution process becomes quite slow or even stops.

3.2. Grey System Theory
Grey systems theory is one of the new mathematical theories effectively used to solve
uncertainty problems with discrete data, incomplete samples and imprecise information
(Tseng, 2008). The darkness of colors is generally used to display the degree of information
clarification. Accordingly, in grey systems theory all systems are categorized into three white,
grey, and black groups. The white parts show deterministic and clear information in a system,
while the black part has completely uncertain characteristics (Deng, 1984). Hence, the grey
parts inherently include insufficient information which stays between two sharp boundaries.
Therefore, a grey number G can be specified by a closed interval including upper and lower
limits [ ( ), ( )]GG
x x  . This method fulfils the expression of uncertain information when it is
difficult to identify the probability density and the membership functions (Chang et al., 1996).
The definition of Grey number is as follows:
Let X be the universal set. Then a grey set G of X is defined by its two mappings as
[ ( ), ( )]GG
G x x   in which ( ) : [0,1]G x x  and ( ) : [0,1]G
x x 
and ( ) ( )GG
x x  , x X . ( )G x and ( )G
x are the upper and lower membership
functions in G respectively. When ( ) ( )G G
x x  , the grey set G becomes a fuzzy set. It
shows that grey theory considers the condition of the fuzziness and can deal flexibly with the
fuzziness situation.
The operations on Real numbers can be extended to Grey numbers which are defined
on sets of Intervals (Moore, 1966). Suppose Grey numbers 1 1 1[ , ]G G G  and 2 2 2[ , ]G G G 
the basic operations on grey numbers are defined as follows (Wu et al., 2005):
1 21 21 2 ,G G G G G G       
2 11 21 2 ,G G G G G G       
   2 1 1 2 2 1 1 21 2 1 2 1 2 1 21 2 min , , , ,max , , ,G G G G G G G G G G G G G G G G G G   
 
111 2
22
1 1
, ,G G G G
G G
 
       
 
 ., 21 GkGkk 
3.2.1. Finding the weights of attributes
Grey theory can effectively be utilized in order to calculate the attribute weights of
incomplete information gathered from subjective judgments of DMs (decision makers). This
procedure that can be employed instead of pair-wise comparisons (commonly used in
Analytic Hierarchy Process) at least decreases the number of questions must be answered by
DMs. Therefore, considering a large number of qualitative attributes the grey based
techniques are more efficient in comparison with the traditional MADM techniques such as
AHP (Dabbaghi et al., 2010).
3.2.2. Definition of Grey Linguistic Variables
The linguistic variables are used to represent imprecision of data and DMs’
preferences over the attributes in the evaluation process. In this research, the attribute weights
are considered as linguistic variables. These linguistic variables can be expressed in grey
numbers by the 1–7 scales, concerning Dabbaghi et al. (2010), as shown in Table 2.

Table 2
The scale of attribute weights.
VH (very
high)
H (high)
MH (middle
high)
M
(middle)
ML (middle
low)
L (low)
VL (very
low)
Scale
[0.9,1.0][0.7,0.9][0.6,0.7][0.4,0.6][0.3,0.4][0.1,0.3][0.01,0.1]W
3.2.2. General Procedure of Finding Weights
Assume that  1 2, , , nQ Q Q Q is a set of n attributes of FMS evaluation and the
attributes are additively independent.  1 2, , , nw w w w     is the vector of attribute
weights. Suppose that the decision group consists of K persons. According to Geometrical
Mean for Grey numbers, the weight of the jth attribute can be calculated as:
1 2 kk
j j j jw w w w     
Where ( 1,2, , )k
jw j n  is the subjective judgment of the kth DM over the jth
attribute described by grey numbers [ , ]k k k
j j jw w w  , where 0k
jw  .
Focusing on the most important attributes facilitates the evaluation process. In order to
remove the attributes, those are not essentially important; we suggest and employ the
following definition.
3.3. Goal Programming (GP)
Goal Programming (Lee, 1972) has been widely applied to solve different real-world
problems which involve multiple objectives. Goal programming (GP) is designed to deal with
problems involving multiple conflicting objectives. However, to overcome the drawback of
GP, decision-makers must specify their goals and priorities beforehand. A systematic
procedure is needed to determine the following factors in constructing the GP model through
group discussion: (1) objectives, (2) desired level of attainment for each objective, (3) degree
of interdependent relationships, and (4) penalty weights for over achievement or
underachievement of each goal (Chang et al., 2009). The optimal configuration of decision
variables (parameters) minimizes the sum of weighted penalties subject to a series of
technological and managerial constraints (Azadeh et al., 2010).
4. Proposed Approach
In order to solve the FMSs combination problem, we proposed a genetic algorithm
based grey system theory approach to solve this multi objective selection problem. On
occasions, we not only face the FMS selection problems for a factory, as solved by many
researchers, but also we may face with multi FMSs combination selection problem. In a multi
FMSs combination problem, each FMS consists of several alternatives. In order to find the
optimum FMSs combination, we have to consider many solutions and in some cases the
problem becomes NP-hard. The more the problem size increases, the more difficult the
problem becomes. In this case, using a well-known Meta heuristic algorithm, genetic
algorithm (GA), is applicable. GAs are successful in solving such problems. Also, the
formulated problem is very difficult to be solved if the factory has a structure with a large
number of FMSs, and thus it is troublesome to be solved by linear programming software,
such as Lingo, as the selection problem includes 0-1 flexibility relations. In addition, the
problem should be formulated. Since the problem is based on multi objectives (cost, yield,
WIP, tardy jobs, etc.) and their priorities are also important to be considered, we applied goal
programming to minimize the total deviation of each objective from its threshold value and to
consider the objectives’ priorities. In addition, number of calculations will be reduced for that
the variables do not need to be converted to the same scale. Because preference information

on attributes of flexible manufacturing systems and on criteria belongs to the DMs’ subjective
judgments cannot be estimated by an exact numerical value, the grey systems theory is
employed to effectively deal with the recognitive uncertainty environment in the FMSs
combination selection problem. Utilizing grey systems theory does not require assuming any
membership function or distribution. In addition, the priorities on both qualitative and
quantitative objectives are considered by employing grey numbers. Moreover, the presented
approach, which introduces integration of the grey systems theory to the goal programming
and genetic algorithm, effectively works with the grey numbers in the entire evaluation and
optimization process and does not require any kind of transformation from grey numbers to
crisp values. The steps of the proposed approach are defined as follows:
Step 1: identify the J objective and subjective attributes (indices) to be considered
(economic, strategic, and operational).
Step 2: constitute a group of I decision makers (DM1, DM2 …DMI) to assess the
subjective attributes of m FMSs each of which has n alternatives utilizing grey numbers and
the geometrical mean as explained in section 3.2.
Step 3: simulate the current shop(s’) situation to obtain the values of non-measurable
operational (quantitative) indices when substituting the new FMSs’ alternatives in the shop.
Turn the obtained valued in to a grey number by adding and subtracting the standard deviation
( ) from the arithmetic mean (  ) to construct an interval for each value[ , ]     .
Step 4: find the J attributes’ preference weights using grey system theory.
Step 5: Minimize the total deviation of attributes from their threshold values using
goal programming with the application of grey numbers. This step is the mathematical
formulation of the attributes.
Step 6: apply the genetic algorithm (GA) to solve the multi FMSs combination
problem using the mathematical formulation in step4 as the fitness function to find the proper
combination of FMSs.
In all steps, the incomplete information of attributes is expressed using grey numbers
as described in section 3.2.
5. An Illustrative Example
In this section, we apply the proposed approach to a practical case using the described
steps in order to find a proper FMSs combination. Figure 1 shows the steps schematically in
brief.
Step 1. in this example, we first identified and then applied the following indices for the
evaluation under study:
- Operational indices: yield which is the throughput minus scrap and rework (Shang and
Sueyoshi, 1995), percent decreased in work in process (WIP), percent decreased in the
number of tardy jobs are taken in to calculations as operational (objective) attributes.
- Strategic indices include space required for each FMS, reliability of the selected FMSs
combination, percent of improvement in quality and flexibility.
- Flexibility is assumed as a significant factor to make the relation of the FMSs in each
combination. Flexibility includes system flexibility, volume flexibility, expansion
flexibility, routing flexibility, process flexibility and product flexibility.
- Economical indices: we applied cost including both initial investment and average
maintenance costs.

Fig. 1. The schematic diagram of the proposed approach.
Step 2. A group of four decision makers (DM) from the top managers are constructed
to evaluate the 4 FMSs that each of them is assumed to have 5 alternatives. Totally, 20 FMSs’
alternatives are assessed by the selected experts to make the best decision for replacing the
current job shop with new FMSs. Each DM should estimate the linguistic judgment (VL, L,
M, etc.) of qualitative attributes utilizing grey system theory. Each linguistic expression
assigned by a DM is turned in to an interval base on Table 2. Then, the interval based
geometrical mean of the 4 DMs’ decisions on one index is assumed as its value for entering in
to evaluation. Table 3 shows a typical example of such judgment for FMS1’s alternatives. For
instance, the first decision maker’s (DM1) opinion about the quality index of FMS11 (the 1st
alternative of FMS1) is a fair quality for that in the grey systems based on Table 2, it is a
number between 4 and 6. Other decisions also follow the same approach. Another subjective
attribute pertains to flexibility. We consider flexibility as the relation of the FMSs two by two
in a selected combination with regard to their volume flexibilities assessing the related value
that how much flexibility relation do they have when, for instance, one of them increases in
volume or changes the product. Before applying judgments, we have to demonstrate the
relation matrix between the four FMSs. Table 4 indicates the flexibility relation matrix of the
four FMSs using 0-1 matrix. 0 depicts no relation and 1 shows that there is a flexibility
relation. The same evaluation process occurs for flexibility as well as quality index. Having
the flexibility relation with respect to Table 4, the FMSs’ alternatives are compared to obtain
their subjective flexibility relation values.
Table 3
An illustrative example of expert linguistic judgments for quality index values of FMS1 alternatives.
Experts FMS11 FMS12 FMS13 FMS14 FMS15
DM1 F MG P F VG
DM2 MP F MP G G
DM3 MG G MP G VG
DM4
DM1
G G P MG G
4 6 6 7 1 3 4 6 9 10
DM2 3 4 4 6 3 4 7 9 7 9
DM3 6 7 7 9 3 4 7 9 9 10
DM4 7 9 7 9 1 3 6 7 7 9
Average 4.74 6.24 5.86 7.64 1.73 3.46 5.86 7.64 7.94 9.49
Database
Step 2:
Constructing a group of
DMs for qualitative
attributes
Step 1:
Defining the attributes
Step 3:
Defining quantitative
attributes
Step 4:
Defining the weights of
attributes
Step 5:
Mathematical modeling
Step 6:
Applying GA for solving
the model
Optimum selection

Table 4
Flexibility relation matrix.
FMS1 FMS2 FMS3 FMS4
FMS1 0 1 0 1
FMS2 1 0 1 0
FMS3 0 1 0 1
FMS4 1 0 1 0
Table 5 illustrates an example of the calculation of flexibility relation values between
FMS1 alternatives and FMS2 alternatives. All the relation values are expressed in grey
numbers and collected in Table 6. The calculated values of all judgments for both the
qualitative and quantitative factors, are collected in Table 7.
Step 3. After calculation of subjective attributes, it is time to find the values of
quantitative factors. It is simple to find some of these attributes such as space required and
reliability, which are defined through the information obtained from the FMSs suppliers. We
also suppose for the cost factor, the initial investment and the average maintenance cost for
the period of 10 years as calculated in (Saidi Mehrabad and Anvari, 2009). The cost values
are turned in to grey numbers as the cost estimation is not crisp. Since finding the real
changes in the number of tardy jobs, WIP and yield are difficult after replacing the new FMSs
combination with the existing equipments, these values should be extracted by simulation.
Thus, percent decrease in the number of tardy jobs, percent decrease in WIP, and Yield are
assumed to be obtained from simulation study presented by Saidi Mehrabad and Anvari
(2009) and Shang and Sueyoshi (1995). The attained values are turned in to grey numbers to
deal with their uncertainty as illustrated in Table 7. Each FMS is assumed to have five
alternatives to be selected by DMs.
Step 4. In order to find the weights of indices, we utilize the grey system theory along
with the expert judgments based on the explanation presented in section 3.2. Each expert
indicates his judgment for each alternative’s importance using the subjective statements (VL,
L, etc.) in Table 2. The obtained weights indicate the priorities of the objectives and will be
used in the next steps as wi in the minimization function of the goal programming model.
Table 8 represents the weights all in grey numbers.
Step 5. We constructed a mathematical model for FMSs combination selection
problem in which cost, Yield, WIP, number of tardy jobs, required space, quality, reliability
and flexibility are taken into consideration. The notations used to develop the mathematical
model for this selection problem are as follows.
Parameters of the model and notation
Since GA will be applied for solving the selection problem, the indices for each FMS,
alternatives of the FMSs should be identified. Additionally, some of the other notations for
the specified attributes should also be defined as grey variables:
Table 5
An illustrative example of expert’s linguistic judgments of DMs for flexibility relation values between FMS1 and
FMS2 alternatives.
Experts FMS11-FMS21 FMS11-FMS22 FMS11-FMS23 FMS11-FMS24 FMS11-FMS25
DM1 VG MG G VG G
DM2 F F F G F
DM3 G G MG F G
DM4 VG G VG G F
DM1 9 10 6 7 7 9 9 10 7 9
DM2 4 6 4 6 4 6 7 9 4 6
DM3 7 9 7 9 6 7 4 6 7 9
DM4 9 10 7 9 9 10 7 9 4 6
AVERAGE 6.90 8.57 5.86 7.64 6.24 7.84 6.48 8.35 5.29 7.35

, :i FMS indices, 1,2,3,..., , 1,2,3,...,i I L 
, :j k FMS’s alternative index, 1,2,3,..., , 1,2,3,...,j J k K 
, :I L Total number of FMSs
, :iJ K Total number of FMSs alternatives available for FMS ,i
, ijijijC C C    
: Initial investment together with maintenance cost for alternative j of iFMS
[ , ]ij ijij
Q Q Q  : Percent improvement in Quality by alternative j of iFMS
[ , ]ijijijWIP WIP WIP  : Percent improvement in alternative j of iFMS
[ , ]ijijijYD YD YD  : yield of alternative j of iFMS when replacing with old equipments
[ , ]ijijijTJ TJ TJ  : Percent decrease in number of tardy jobs of alternative j of iFMS
jk
ilFLEX [ , ]
jk jk
il ilFLEX FLEX Flexibility relation value for alternatives ,j k of FMSs ,i
ijREL : Reliability for alternative j of iFMS
ijSR : Space required for alternative j of iFMS
TC : Total costs of an FMSs combination provided by the selected alternatives,
[ , ]:TQ TQ TQ  Total Quality improvement of an FMSs combination provided by the
selected alternatives,
[ , ]:TYD TYD TYD  Total yield of an FMSs combination provided by the selected
alternatives,
[ , ]:TTJ TTJ TTJ  Total decrease in number of tardy jobs of an FMSs combination
provided by the selected alternatives,
[ , ]:TFLEX TFLEX TFLEX  Total flexibility of the selected FMSs,
[ , ]:TWIP TWIP TWIP  Total decrease in WIP of the selected FMSs,
[ , ]:TTJ TTJ TTJ  Total decrease in number of tardy jobs of the selected FMSs,
TREL : Total Reliability of an FMSs combination provided by the selected alternatives,
ijSR : Space required for alternative j of iFMS .
The goal programming can be mathematically formulated as follows: The model
results in finding the optimum FMSs combination.
Objective function
Equation (1) shows the objective function minimizing the total deviation of the
objectives (Cost, Yield, WIP, etc.) as the objectives pertain to the FMSs’ alternatives from
their acceptable thresholds. (di is deviation from desired level of the objectives and i is the
number of objectives). wi are the weights obtained in Step 4:
8
1 i ii
Minimize Z
Z w d

   
(1)
Constraints
Equation (2) shows the total cost as a grey number. The total cost is the sum of initial
investment and average maintenance of all FMS’s alternative in a selected combination:
1 1
( ),iI J
ij iji j
TC C Y i l 
      (2)

Equation (3): Total flexibility relation value as a grey number. Displays the total
flexibility relationships between the FMSs’ alternatives for a selected FMS combination:
1 1 1 1
( ),i lI J l k jk jk
il il ij ilki j l k
TFLEX FLEX X Y Y i l   
          (3)
Equation (4) shows the total decrease in WIP as a grey number. The total decrease in
WIP is the sum of this value of FMSs from all selected alternatives:
Equation (5). Total tardy job is the total percent decrease in the number of tardy jobs
in the existing job shop’s products when replacing with new FMSs combination:
Equation (6). Total yield is the sum of yield for a selected FMSs combination:
1 1
( )
i
ij ij
i j
JI
TYD YD Y
 
    (6)
Equation (7). Total quality is the sum of improvement in the quality level of the
existing job shop’s products when replacing with new FMSs combination. This qualitative
factor is considered as a grey number to deal with its uncertain information:
Equation (8). Total reliability is the multiple of the reliability (REL) of FMSs from all
alternatives in a selected combination as they assumed to be a series system:
1
( )
I
ij ij
i
TREL REL Y

  (8)
Equation (9). Total space required is the sum of the space each FMS required in a
combination:
1 1
( )
i
ij ij
i j
JI
TSR SR Y
 
  (9)
Constraints (10)-(17) indicate the deviation of the goals from the threshold values
which is minimized through the objective function (1). The right hand side values are
obtained by experts’ judgment:
1 [500 530]TC d     (10)
2 [90 100]TFLEX d     (11)
3 [60 80]TWIP d     (12)
4 [60 80]TTJ d     (13)
5 [95 100]TYD d     (14)
6 [50 60]TQ d     (15)
7 1TREL d  (16)
8 11000TSR d  (17)
i l1,Alternative j of FMS has flexibility relationwith FMS of alternative k
0,
jk
ilX
otherwise

 

(18)
i1,Alternative j of FMS is selected
for all i, j
0,
ijY
otherwise

 

(19)
i1,Alternative j of FMS is selected
for all l, k
0,
lkY
otherwise

 

(20)
Constraints (21), (22) show that only one alternative can be selected for each FMS:
1
1 , ,
i
ij
j
J
Y for all i j

 (21)
1
1 , ,
k
K
kY for all k

 (22)
1 1
( )
i
ij ij
i j
JI
TWIP WIP Y
 
    (4)
1 1
( ),iI J
ij iji j
TTJ TJ Y i l 
      (5)
1 1
( ),iI J
ij iji j
TQ Q Y i l 
      (7)

Table 6
Values of flexibility relation matrix between FMSs’ alternatives.
FMS21 FMS22 FMS23 FMS24 FMS25 FMS31 FMS32 FMS33 FMS34 FMS35 FMS41 FMS42 FMS43 FMS44 FMS45
FMS11[6.9-8.57] [5.85-7.63] [6.23-7.48] [6.48-8.34][5.29-7.34] [7.17-8.67] [0.14-2.05] [6.9-8.14] [7.63-8.9] [5.63-7.54]
FMS12[1.32-3.22] [3.83-5.09] [5.63-7.17] [0.13-1.86][7.17-8.68] [5.86-7.64] [7.17-8.68] [5.86-7.64] [2.45-4.12] [0.03-1.32]
FMS13[1.73-3.46] [5.86-7.64] [1.86-3.83] [2.63-4.56][7.45-9.24] [5.85-7.63] [6.73-8.45] [7.94-9.48] [2.45-4.12] [1.86-3.83]
FMS14[0.59-2.91] [6.24-7.84] [3.72-5.42] [5.09-6.9] [8.13-9.15] [4.9-6.48] [6.09-8.13] [0.42-2.45] [2.45-4.12] 5.63-7.17]
FMS15[6.42-7.75] [6.48-8.35] [2.21-4.41] [4.28-6] [1.86-3.83] [0.32-2.28] [4.9-6.48] [7.94-9.49] [4-6] [1.32-3.22]
FMS21 [4.92-6.64] [7.63-8.9] [0.54-2.63] [3.02-5.04] [0.31-2.27]
FMS22 [6.64-8.05] [5.45-6.9] [0.42-2.45] [4.28-6] [0.17-2]
FMS23 [2-4.24] [8.45-9.74] [2.45-4.12] [6.24-7.45] [1.73-3.46]
FMS24 [6.24-7.45] [1.86-3.83] [0.17-2] [1.32-3.22] [4-6]
FMS25 [0.13-1.86][5.63-7.17] [7.17-8.68] [6.64-7.65] [2.28-3.72]
FMS31 [2.45-4.12][0.42-2.45] [6.48-7.94] [4.43-6.24] [6.24-7.84]
FMS32 [2.45-4.12][0.55-2.63] [6.48-7.94] [4.43-6.24] [5.09-6.9]
FMS33 [2.45-4.12] [0.59-2.91] [6.48-7.94] [4.43-6.24] [4.9-6.48]
FMS34 [2.45-4.12] [0.42-2.45] [6.48-7.94] [4.43-6.24] [5.86-7.64]
FMS35 [2.45-4.12] [0.42-2.45] [6.48-7.94] [5.09-6.9] [4.43-6.24]

Table 7
The qualitative and quantitative values of FMSs’ alternatives.
FMSs alternatives cost ($0000) yield (00) WIP (%) Tardy jobs (%) quality
space
required
(ft2)
Reliability
FMS1
FMS11 [115.3-118.63] [19.6-21.4] [10.1-12.4] [4.8-5.5] [4.73-6.24] 4500 0.94
FMS12 [125.54-128.2] [18.5-20] [4.2-5.8] [8.5-9.8] [5.85-7.64] 5100 0.95
FMS13 [110. 72-112.47] [23.5-25.3] [7.8-9.1] [13-14.5] [1.73-3.46] 5000 0.98
FMS14 [140.36-142.75] [15.4-16.9] [8.4-10] [15-16.30] [5.85-7.64] 3800 0.9
FMS15 [111.9-113.68] [10-12.4] [6.4-9.3] [6.8-7.4] [7.93-9.49] 7100 0.98
FMS2
FMS21 90.82-93.25 [29.5-32.4] [18.1-20.6] [8.89.20] [3.03-5.05] 3400 0.93
FMS22 [88.34-90] [24.5-26.7] [10.3-12.4] [13.8-14.20] [4.74-6.24] 2700 0.9
FMS23 [95-110.42] [28.6-31.2] [9.7-11.2] [0-1.10] [0.41-2.45] 4600 0.98
FMS24 [105.3-107.4] [18.3-21.1] [19.4-20.2] [9.4-10.30] [3.72-5.42] 3000 0.99
FMS25 [95.476-98] [23.5-25.9] [7.8-8.5] [6.4-6.80] [7.63-8.91] 6500 0.94
FMS3
FMS31 [270-272] [15.4-17.7] [5.3-7.8] [5.3-6.70] [7.17-8.68] 2000 0.99
FMS32 [293.35-296] [13.1-14.9] [7.1-9.5] [13.1-14.5] [5.09-6.90] 4500 0.98
FMS33 [261.36-262.7] [18.1-20.3] [10.1-11.3] [2.6-3.80] [0.42-2.45] 3200 0.94
FMS34 [275.45-277.68] [14.5-16.2] [8.3-9.8] [8.4-9.70] [3.72-5.42] 6000 0.96
FMS35 [298.47-300.9] [10-13.4] [4.8-6.9] [4.9-10.50] [7.637-8.91] 5000 0.95
FMS4
FMS41 [62.36-64.8] [11-12.3] [12.6-13.8] [16.2-17.60] [3.56-5.38] 7000 0.93
FMS42 [65.34-67.45] [15-17] [9.5-10.9] [14.6-15.80] [5.09-6.90] 5900 0.9
FMS43 [51.9-53] [20.1-22] [11.4-13.1] [19-20.10] [0.54-2.63] 6100 0.98
FMS44 [50.67-52.19] [13.3-15.1] [8.9-10.1] [9.8-10.70] [3.72-5.42] 3800 0.99
FMS45 [70.872.56] [16.7-19.1] [7.8-9.8] [10.8-11.90] [7.63-8.91] 3200 0.99

Table 8
Priority weights of the attributes or objectives.
Weights
of goals
cost ($0000) yield (00) WIP (%)
Tardy jobs
(%)
space required
(ft2)
Reliability quality Flexibility
[0.845-0.97] [0.90-1] [0.49-0.66] [0.53-0.73] [0.76-0.89] [0.65-0.83] [0.59-0.76] [0.845-0.97]
The above assessment formulated model is very difficult to solve if the FMSs has a structure
with a large number of alternatives. For this reason, a genetic algorithm based approach is
developed to process this assessment model through the illustrated evaluation tables.
Step 6. After collecting data for multi FMSs selection, the genetic algorithm is applied to
solve the proposed mathematical model. The collected data are inserted in-to the mathematical
model and the model works as an evaluation function in the GA. We used MATLAB R2008
software to apply the real coding to find the proper solution for FMSs combination selection
problem. The general steps of GA we followed in this study are stated below:
(1) Generate initial population
Each chromosome in the initial population represents a possible FMSs combination in the
factory. Each gene indicates a possible alternative for an FMS. We assume to have 4 FMSs and
each FMS has 5 alternatives. Thus, there should be 4 genes accordingly in each chromosome and
its value can change in the range of 1-5. The following chromosome is an example of a possible
solution for multi FMS selection. The chromosome shows that alternatives 3, 1, 5 and 3 are
selected for each FMSs of 1-4 respectively. In this study, we tested various numbers of initial
populations.
FMS items FMS1 FMS2 FMS3 FMS4
Alternatives FMS13 FMS21 FMS35 FMS43
(2) Evaluate fitness function of each chromosome
For each chromosome, a grey number ( Z ), representing the total deviation of each FMSs
combination from the ideal values based on the mathematical model, is calculated as the fitness
function. All the equations 3-24 are applied in the computation. Each chromosome that does not
meet all the constraints will not be allowed to enter in to the population. Hence, the pop will
remain feasible.
(3) Select 4 random chromosomes from population for tournament selection.
In order to select a parent for cross over and mutation, we need to select them from the
population pool. We applied the tournament selection. 4 chromosomes are selected they should
be compared with each other two by two. The two chromosomes with minimum deviations (Z)
will be selected as the parents that cross over should be applied on.
(4) Apply cross over to the selected parents to produce new off spring.
Uniform cross over is adopted for crossing the selected parents (two chromosomes in each
generation) in order to generate a new offspring. The procedure is that each gene of the two
selected parents is compared one by one. If they are identical, the same alternative will be settled
in that gene. If they differ, one of the FMSs’ alternatives will be chosen by the probability of 0.5.
The process goes on until reaching the end of the chromosomes. Figure 2 illustrates the cross
over procedure.

The first parent
The second parent
The new off spring
after crossover
Fig. 2. Uniform crossover
(5) Apply mutation to the new offspring
The new offspring is then ready for mutation. Single point mutation is adopted. First, a
number between 1 and 4 is randomly selected. Then, the randomly selected gene is mutated
based on the FMSs’ number of alternatives range of the selected gene (1-5). A random number in
that range is generated to be replaced by the original number. Figure 3 shows the mutation
procedure. Assume that 3 is the selected random number shown by an arrow. This FMS has 5
alternatives. Thus, a random number between 1 and 5 is then selected to replace this alternative.
If 2 is selected 5 will be replaced by 2. It is notable that mutation is applied on the chromosome
selected from tournament selection.
Before mutation
After mutation
Fig. 3. Single point mutation
(6) Enter the new offspring to the population utilizing replacement strategy
The new offspring is replaced with the worst chromosome (the maximum fitness, Z) in the
population that is sorted based on their Z values.
(7) Stopping condition
Stopping condition used in this study is based on the number of generations which can be
randomly generated from the system interface. Thus, the initial setting and the number of
generations must be entered before running the genetic algorithm. Termination occurs when the
number of iterations reaches the set number of generations.
We set 100, 200, 300 and 400 as the number of generation in the program together with the
different initial populations; 20, 50. Satisfying the stopping condition, the GA procedure will be
continually repeated. Figures 4–7 illustrate the comparable results of separate GA
implementation based on the generations of 100, 200, 300, 400 and the pop size of 50. GA
converges to the interval of Z = [680.8-862.89] as a grey number for the generations of 200
and above it. For generation 100 the obtained intervals of the objective function values were in a
higher range of total deviation. Thus, for this case we found the 4 2 1 5 as a suitable combination
for FMSs. Which means alternative 4 is selected for FMS1, alternative 2 for FMS2, alternative 1

for FMS3 and alternative 5 is selected for FMS4. The achieved interval for Z, as a grey number,
facilitates decision making for the managers as they can find other combinations for this
selection problem provided that the results lie in to this interval.
Fig. 4. GA convergence to the interval of [z1, z2]- generation 100-popsize 50
Fig. 5. GA convergence to the interval of [z1, z2]- generation 200-popsize 50

Fig. 6. GA convergence to the interval of [z1, z2] - generation 300-popsize 50
Fig.7. GA convergence to the interval of [z1, z2]- generation 400-popsize 50
6. Conclusions and Further Work
This paper considers a new problem for sustainable flexible manufacturing systems
selection and evaluation. All previous works provided numerous solutions to the problem of
FMS selection when having one FMS with different alternatives. But we raised a new problem of

FMS decision making when there are more than one FMS to be selected in a manufacturing
enterprise. Thus, a proper combination of FMSs should be selected. Each of these FMSs has
several alternatives with the flexibility relations with other FMSs’ alternatives. Therefore, we
develop a goal programming to formulate this problem. The subjective data were obtained by the
experts’ judgments based grey system theory to deal with the incomplete information of the
qualitative attributes. Then a GA based model is adopted to solve this difficult decision making
problem. An illustrative example adopted from a real automobile manufacturing (with some
changes in data) is developed to test the validity of the proposed model. A group of four decision
makers (DM) were constructed to evaluate the 4 FMSs that each one is assumed to have 5
alternatives. The problem can be solved easily for each alternative separately if the relation
between the alternatives is not considered. But there is flexibility relation between alternatives
that makes the problem solving more difficult than usual. Consequently, GA was used to solve
the problem easily. The outputs of the proposed approach presented the nearly best alternative
for each of the FMSs evaluated based on both objective and subjective attributes simultaneously.
That means, the 4th
, 2nd
, 1st
, 5th
alternatives of FMS 1–4 were selected as a suitable combination
for the FMSs under study.
The results show that by means of this approach, we can effectively offer a faster overall
evaluation speed, moreover, the utilization of the grey system theory can put the results in
intervals facilitating decision making as the manager can change the adopted result to other
possible combinations, based on their experience, provided that the solution lie in to the achieved
optimal interval. Additionally, form the sustainability point of view, the proposed approach helps
manufacturers to purchase flexible systems considering the systems relations with each other and
to manufacture the products in a flexible way with a better quality, less costs and less reworks
than before. Future research can include an extension of the developed approach to more
complex problems such as combination problems involving FMSs sequence consideration.
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